TELESCOPE. 



diTifor of the crown; and 7,^^ 



l.OOO 



= -5735 X 2 = 



1.147, the divifor of the flint. In the next place we have 



'•°°° = .94697 for the refrafted focus of the crown lens ; 



1.056 



and becaufe the foci of the crown and flint lenfes mud be in 



the fame ratio as their difperfive powers, which we have 



Hated to be I : 1.524. ^^^ ^^^11 have ^^ = i.3827forthe 



1. 147 



rcfraftcd focus of the concave or flint-glafs. Now, having 

 .94697 : 1.3827 as the ratio of the two feparate re- 

 frafted focal diftances that fhall banifh all colours by their 

 equal and oppofite difperfive powers ; we next find what will 

 be the compound focus correfponding to thefe two when 

 put in contaft. Let F be the focQs of the convex, and 

 'F that of the concave ; and by our /raff/ca/ theorem 5. there 



F X 'F 



will be ;= = = the compound focus, which 



'F — F 



pla 



numbers will ftand thus ; 



■94697 X 



1.3827 



.94697 



the 



1.3827 



proportional compound focus required. Now if the prifma- 

 tic aberration were the only one neceffary to be counterafted, 

 we have already obtained numbers that would enable us to con- 

 ftruft an achromatic or colourlefs compound objed-glafs ; for 



30 



without interfering with the remedy which we have juft pre- 

 fcribed for the other. In order to mark the diftinftion that 

 muft be made in the fymbols, as applied refpeClively to the 

 convex and concave lenfes, let it be underftood, that the fub- 

 joined notation will be attended to in our inveftigation of the 

 curves proper for our prefent purpofe ; viz. 



means the radius of the firft furface. 



means the radius of the fecond furface. 



means the focus from folar rays, or geome- 

 trical, if fo exprefled. 



the thicknefs of the lens. 



the fpherical aberration. 



the fine of incidence. 

 « 'n the line of refraftion. 



* the compound focus. 



It may be alfo neceflary to premife, that whatever ratio 

 of the radii r and R be fixed upon for the convex lens, the 

 ratio V : 'R of the concave may always be found by proper 

 inveftigation fuch, that its aberration will countcraiSt that 

 of /■ : R ; but the reverfe is not true ; the aberration of 

 'r : 'R may be too great for the aberration of any ratio 

 r : R to equal ; therefore the ratio r : R is firft alTumed, 

 as is moft convenient for the optician's tools already 

 formed ; and 'r : 'R muft be fo calculated, that its aberration 

 ftiall be in due proportion for correAing the aberration of 

 the afTumed convex lens. We now have to do with the geo- 

 metrical foci of both lenfes, when their radii become the 



if we multiply F, 'F, and / alike by -^^, or 9.12, the fubjeft of inveftigation ; and we have feen that 9.12 I- j 



geometrical focus of the convex lens, we fliould have the 

 abfolute refr afted focus of F — .94697 x 9.12]= 8.636 ; 

 that of 'F = 1.3827 X 9.12I = 12.61 ; and the compound 

 focus = 3.29 X 9.12^ = 30 very nearly ; and it would be 

 immaterial what the curves were, provided the refratled 

 focal diftances of F and 'F were as above ftated ; but as 

 the tools for forming the curves refpeftively tor the fides of 

 thefe lenfes, muft have regard to the radii of curvature, it 

 would be now neceflai'y to ufe the divifors as multipliers, to 

 convert the rjfrac5ted into the geometrical foci, and then the 

 bufinefs might be put in hand. On this fuppofition, of there 

 being only one kind of aberration, the conftruftion of a 

 compound achromatic objeft-glafs would be no difficult 

 aff"air ; for while the focal diftances only are required to be 

 to each other in a given ratio, the radii of curvature might 

 be varied almoft at pleafure, without affefting the focal 

 diftance. But there yet remains the fpherical aberra- 

 tions of the two feparate lenfes to oppofe to each other in 

 fuch proportion, that their tendency to produce indiJlincJnefs 

 may be completely obviated. Before the time of fir Ifaac 

 Newton, this was the only kind of aberration that opticians 

 thought they had to contend with ; and though it ie fmall 

 in quantity, compared to the prifmatic aberration, yet it is 

 more difficult to conquer. It is, however, contrary to the 

 opinion of that great philofopher and mathematician, in the 

 power of the modern optician to cure this defeft of fpheri- 

 cal glafles, by means equally fimple, when determined, as 

 thofe by which the prifmatic colours are made very nearly to 

 vanifti. As in the annihilation of the prifmatic colours, the 

 ratio of the focal diftances, made direftly as the ratio of the 

 difperfive powers, is a cure for the firft imperfeftion ; fo the 

 ratio of the radii, r : R, of the two lenfes, fo calculated as to 

 counteraft each other's fpherical aberrations, is the cure for 

 the fecond imperfection ; and this cure we have yet to apply, 



10 



is the geometrical focus of the convex lens, therefore 

 1.524 X 9.12 = 13.9 is the geometrical focus of the eon- 

 cave, their ratio being ftill as their difperfive powers, 

 very nearly. Let us now aflume r = 7.5, or any other 

 quantity at option, and fee by the proper theorem what R 

 will be, to have a focus of 9. 1 2 inches : to do this we have, by 



No. I . of our praSical tl>eorems, before given, 



R, 



or, in figures. 



IS X 9-12 



2 X 



= R 



1 1.63 ; hence 



7.5 - 9.12 



)• : R :: 7.5 : 11.63, o"" ^^ ^ • I-55- I" ^^^ "^"^ place, we 

 muft determine what is the longitudinal aben-ation arlfing 

 from the figure of a lens, where the ratio r : R is 1 : 1.55, 

 which is moft conveniently done by the general theorem of 

 Huygens, which we have before exemplified, and which 



ftandsthus; ^-l^^L^^^i^' X T = A = 1.3614 x T. 



^^ = 



r-i-Ri'= 1 + 1-55 

 multiplied by 

 value of deno-1 



rmnator 



6.5025 

 _6 



39-015 



Then 



53-1175 



.3614 X T = A. 

 39.015 



Having now found 1.361 X T = A of the convex lens, 

 the value of T, which is the fum of the verfed fines of the 

 two interfering curves of its furfaces, may be calculated by 

 the fquare root, or by plane trigonometry, and will be 

 found = .252, when the femi-diameter of the lens is 1.5, 

 confequently 1.361 x .252 = .3429, is the abfolute quantity 

 of the fpherical aberration of the convex lens ; but 'T of the 



concave 



